Finite Group Databasebeta

← #116868

Subgroups — #116868

Order 512.

621 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 621 conjugacy classes of subgroups — too many to draw. Only its 124 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20717C211yesmaximal, characteristic
2#17815C211yesmaximal, characteristic, complemented
3#17357C212yesmaximal
4#8000C211yesmaximal, characteristic, complemented
5#7970C212yesmaximal, complemented
6C22×#117C2211yescharacteristic
7#2920C2212yes—
8#2920C2212yes—
9C2×#432C2211yescharacteristic
10C2×#431C2211yescharacteristic
11#1817C412yes—
12#1798—22no—
13#1779C412yes—
14#1750—21no—
15#1750—21no—
16#1572—24no—
17#1567—21no—
18#1566—21no—
19#1464—24no—
20#1463—22no—
21#1462—22no—
22#1458—24no—
23#1292—22no—
24#1287—24no—
25C24×C4C2×C411yescharacteristic
26C2×#117—21no—
27C2×#117—22no—
28C2×#117—22no—
29C2×#117—21no—
30C2×#117—21no—
31C2×#117—21no—
32C2×#117—22no—
33C2×#117—22no—
34C22×C4⋊C4D411yescharacteristic
35C22×C4⋊C4Q811yescharacteristic
36C22×C22⋊C4D411yescharacteristic
37C22×C22⋊C4Q811yescharacteristic
38C22×C42C2×C411yescharacteristic
39C22×C42C2311yescharacteristic
40#432—22no—
41#432—22no—
42#432—22no—
43#432—22no—
44#432—22no—
45#432D414yes—
46#431—22no—
47#431—22no—
48#431—22no—
49#431—22no—
50#431—22no—
51#431D414yes—
52#423—22no—
53#423—22no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8D412yes—
56#407—22no—
57#407—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8C2×C412yes—
61C2×C22⋊C8C2×C412yes—
62C4×M16—22no—
63C4×M16—22no—
64C2×#97—21no—
65C2×C4×C8D411yescharacteristic
66C2×C4×C8D411yescharacteristic
67#336—22no—
68#336—22no—
69#335—24no—
70#334—24no—
71#324—28no—
72#323—28no—
73C25C4⋊C411yescharacteristic
74C23×C4—21no—
75C23×C4—22no—
76C23×C4—21no—
77C23×C4—22no—
78C23×C4—44no—
79C23×C4C4⋊C411yescharacteristic
80C23×C4C4⋊C411yescharacteristic
81C23×C4C4⋊C411yescharacteristic
82C23×C4C22×C411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×Q811yescharacteristic
85C2×M16—42no—
86C2×M16—42no—
87C22×C8—22no—
88C22×C8C22⋊C412yes—
89C22×C8C22⋊C412yes—
90#117—41no—
91#117—42no—
92#117—42no—
93#117—42no—
94#117—41no—
95#117—42no—
96#117—42no—
97#117—41no—
98#117—42no—
99#117—41no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—22no—
119C2×C42—22no—
120C2×C42C2×D412yes—
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D412yes—
124C2×C42#4111yescharacteristic
125C2×C42#4112yes—
126C2×C42#4112yes—
127C2×C42#4111yescharacteristic
128C4⋊C8—24no—
129C4⋊C8—24no—
130C4⋊C8—24no—
131C4⋊C8—42no—
132C4⋊C8—42no—
133C4⋊C8—42no—
134C22⋊C8—24no—
135C22⋊C8—24no—
136C22⋊C8—24no—
137C22⋊C8—24no—
138C22⋊C8—44no—
139C22⋊C8—44no—
140C22⋊C8—42no—
141C22⋊C8—42no—
142#97—24no—
143#97—42no—
144#97—41no—
145#97—41no—
146C4×C8—21no—
147C4×C8—21no—
148C4×C8—24no—
149C4×C8—21no—
150C4×C8—21no—
151C4×C8—42no—
152C24—21no—
153C24—22no—
154C24—42no—
155C24—42no—
156C24—81no—
157C24C2×C4⋊C411yescharacteristic
158C22×C4—24no—
159C22×C4—28no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—44no—
171C22×C4—42no—
172C22×C4—44no—
173C22×C4—42no—
174C22×C4—44no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4C2×C22⋊C411yescharacteristic
182C22×C4#11711yescharacteristic
183C22×C4C4×D412yes—
184C22×C4C4×D412yes—
185C22×C4#12012yes—
186C22×C4#12111yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4C22⋊Q812yes—
189C22×C4C22⋊Q811yescharacteristic
190C22×C4C22⋊Q811yescharacteristic
191C22×C4C22⋊Q812yes—
192C22×C4#12312yes—
193M16—84no—
194M16—84no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—44no—
203C2×C8—44no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C42—41no—
217C42—42no—
218C42—41no—
219C42#12011yescharacteristic
220C42#12112yes—
221C42#12112yes—
222C42#12112yes—
223C42#12311yescharacteristic
224C42#12312yes—
225C42#12412yes—
226C42#12411yescharacteristic
227C42#12612yes—
228C42#12711yescharacteristic
229C23—22no—
230C23—22no—
231C23—24no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—41no—
239C23—82no—
240C23—82no—
241C23—88no—
242C23#38011yescharacteristic
243C23#38512yes—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—48no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—44no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4#38611yescharacteristic
271C2×C4#38812yes—
272C2×C4#39011yescharacteristic
273C2×C4#39211yescharacteristic
274C2×C4#39311yescharacteristic
275C2×C4#39412yes—
276C2×C4#39612yes—
277C2×C4#39711yescharacteristic
278C2×C4#39911yescharacteristic
279C8—42no—
280C8—42no—
281C8—82no—
282C8—84no—
283C8—84no—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—44no—
290C22—44no—
291C22—42no—
292C22—82no—
293C22—84no—
294C22—84no—
295C22—81no—
296C22—81no—
297C22#143911yescharacteristic
298C22#181212yes—
299C4—21no—
300C4—22no—
301C4—21no—
302C4—22no—
303C4—21no—
304C4—21no—
305C4—22no—
306C4—21no—
307C4—42no—
308C4—81no—
309C4—81no—
310C4#189112yes—
311C2—22no—
312C2—42no—
313C2—82no—
314C2#1107111yescharacteristic
315C2#1109811yescharacteristic
316C2#1110311yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C4⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #17815C2semidirect
4. #8000C2semidirect
5. #7970C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#207172
#178152
#173572
#80002
#79702